1. # Equivalent definitions for $BMO$

The space of functions of bounded mean oscillation $BMO$ is defined by the BMO norm $$\label{norm} ||f||_{BMO} = \sup_{\text{cubes }Q} \frac{1}{|Q|} \int_Q |u(y) - u_Q| dy$$

But an equivalent definition is to take the sup over balls instead of cubes ...

2. # The space $BMO$
## Introducing $BMO$
The space of functions of bounded mean oscillation or $BMO$ arises when studying the space of functions whos deviation from the mean over cubes is bounded. In ways $BMO$ is similar to $L^\infty$, and it is often used as a replacement, however functions in $BMO$ may be ...